Solution of the Schr\"odinger equation making use of time-dependent constants of motion

TL;DR

The paper demonstrates that solutions to the Schrödinger equation can be constructed using time-dependent constants of motion, specifically by employing common eigenfunctions of commuting operators that are constants of motion.

Contribution

It introduces a method to solve the Schrödinger equation utilizing time-dependent constants of motion and their eigenfunctions, providing a new approach to quantum dynamics.

Findings

Common eigenfunctions of commuting constants of motion solve the Schrödinger equation.

Operators for initial Cartesian coordinates are constants of motion and help derive the Green function.

The method simplifies finding solutions to quantum systems with conserved quantities.

Abstract

It is shown that if a complete set of mutually commuting operators is formed by constants of motion, then, up to a factor that only depends on the time, each common eigenfunction of such operators is a solution of the Schr\"odinger equation. In particular, the operators representing the initial values of the Cartesian coordinates of a particle are constants of motion that commute with each other and from their common eigenfunction one readily obtains the Green function.